Let
$$
\Delta(2,3,7)
=\langle A,B,C\mid A^2=B^3=C^7=ABC=1\rangle
$$
be the orientation-preserving $(2,3,7)$ <hyperbolic triangle group>. Define
$$
a\mapsto A,\qquad b\mapsto B,\qquad c\mapsto C,
\qquad x\mapsto1,\qquad y\mapsto1.
$$
Every defining relator of $G$ maps to the identity, and $A,B,C$ lie in the image, so this is a surjective <group homomorphism>. Since
$$
\frac12+\frac13+\frac17<1,
$$
$\Delta(2,3,7)$ is a non-elementary <Fuchsian group>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 133 / 3 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-133/3/b/solution